Ellipses!

Standard Form: x^2/a^2+y^2/b^2=1
--To put in standard form, either divide or complete the square.

Sketching an Ellipse:

1. Find the major axis (the major axis is the variable with the largest denominator).


2. Find the minor axis (the minor axis is the variable with the smallest denominator).


3. Find the vertex (the vertex is the square root of the largest denominator - to put in point form if the major axis is "x" the answer goes in the "x" spot and vise versa).


4. Find the other intercept (the other intercept is the square root of the smallest denominator, put in point form).


5. Find the length of the major axis (two times the square root of the largest denominator).


6. Find the length of the minor axis (two times the square root of the smallest denominator).


7. To find the foces, vertex, or other intercepts, use the formula: "smallest denominator=largest denominator-focus^2" or "other intercept^2=vertex^2-focus^2".

-- The focus is always on the major axis and is a point.

Example:

(x^2/4)+(y^2/9)=1

1. The "y" axis is the major axis because it has the largest denominator.


2. The "x" axis is the minor axis because it has the smallest denominator.


3. The square root of 9=+/-3, which gives you the vertices of (0,3) and (0,-3).


4. The square root of 4=+/-2, which gives you the other intercepts of (2,0) and (-2,0).


5. Two times the square root of 9(largest denom.)=2(3)=6


6. Two times the square root of 4(smallest denom.)=2(2)=4


7. Focus: 4=9-focus^2

-5=-f^2

Focus= +/- square root of 5

(0, square root of 5) and (0, -square root of 5)